Page 1

arXiv:1009.0304v2 [cs.IT] 28 Feb 2011

Joint Source-Channel Coding with Correlated

Interference

Yu-Chih Huang and Krishna R. Narayanan

Department of Electrical and Computer Engineering

Texas A&M University

{jerry.yc.huang@gmail.com, krn@ece.tamu.edu}

Abstract

In this paper, we study the joint source-channel coding problem of transmitting a discrete-time analog source over an additive

white Gaussian noise (AWGN) channel with interference known at transmitter. We consider the case when the source and the

interference are correlated. We first derive an outer bound on the achievable distortion and then, we propose two joint source-channel

coding schemes to make use of the correlation between the source and the interference. The first scheme is the superposition of

the uncoded signal and a digital part which is the concatenation of a Wyner-Ziv encoder and a dirty paper encoder. In the second

scheme, the digital part is replaced by a hybrid digital and analog scheme so that the proposed scheme can provide graceful

degradation in the presence of (signal-to-noise ratio) SNR mismatch. Interestingly, unlike the independent interference setup, we

show that neither of both schemes outperform the other universally in the presence of SNR mismatch. These coding schemes are

further utilized to obtain the achievable distortion region of the generalized cognitive radio channels.

Index Terms

Distortion region, joint source-channel coding, cognitive radios.

I. INTRODUCTION AND PROBLEM STATEMENT

In this paper, we consider transmitting a length-n i.i.d. zero-mean Gaussian source Vn= (V (1),V (2),...,V (n)) over n

uses of an additive white Gaussian noise (AWGN) channel with noise Zn∼ N(0,N·I) in the presence of Gaussian interference

Snwhich is known at the transmitter as shown in Fig. 1. Throughout the paper, we only focus on the bandwidth-matched

case, i.e., the number of channel uses is equal to the source’s length. The transmitted signal Xn= (X(1),X(2),...,X(n))

is subject to a power constraint

1

n

i=1

where E[·] represents the expectation operation. The received signal Ynis given by

Yn= Xn+ Sn+ Zn.

n

?

E[X(i)2] ≤ P,

(1)

(2)

We are interested in the expected distortion between the source and the estimate?Vnat the output of the decoder given by

where f and g are a pair of source-channel coding encoder and decoder, respectively, and d(.,.) is the mean squared error

(MSE) distortion measure given by

d(v, ˆ v) =1

n

i=1

Here the lower case letters represent realizations of random variables denoted by upper case letters. As in [1], a distortion D

is achievable under power constraint P if for any ε > 0, there exists a source-channel code and a sufficiently large n such that

d ≤ D + ε.

When V and S are uncorrelated, it is known that an optimal quantizer followed by a Costa’s dirty paper coding (DPC) [2]

is optimal and the corresponding joint source-channel coding problem is fully discussed in [3]. However, different from the

d = E[d(Vn,g(f(Vn,Sn) + Sn+ Zn))],

(3)

n

?

(v(i) − ˆ v(i))2.

(4)

ENC

++

DEC

V

ˆV

Z

S

X

Y

Fig. 1. Joint source-channel coding with interference known at transmitter.

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1

typical writing on dirty paper problem, in this paper, we consider the case where the source and the interference are correlated

with a covariance matrix given by

?

Under this assumption, separate source and channel coding using DPC naively may not be a good candidate for encoding

Vnin general. It is due to the fact that the DPC tries to completely avoid the interference without signal to noise ratio (SNR)

penalty so that it cannot take advantage of the it correlation between the source and the interference. In this paper, we first

derive an outer bound on the achievable distortion region and then, we propose two joint source-channel coding schemes which

exploit the correlation between Vnand Sn, thereby outperforming the naive DPC scheme. The first scheme is a superposition

of the uncoded scheme and a digital part formed by a Wyner-Ziv coding [4] followed by a DPC, which we refer to as a

superposition-based scheme with digital DPC (or just the superposition-based scheme). The second scheme is obtained by

replacing the digital part by a hybrid digital and analog (HDA) scheme given in [3] that has been shown to provide graceful

degradation under an SNR mismatch. We then analyze the performance of these two proposed schemes for SNR mismatch

cases. It is shown that both the HDA scheme and the superposition-based digital scheme benefit from a higher SNR; however,

interestingly, their performances are different.

One interesting application of this problem is to derive the achievable distortion region for the generalized cognitive radio

channels considered in [5] (also in [6]). This channel can be modeled as a typical two-user interference channel except that one

of them knows exactly what the other plans to transmit. We can regard the informed user’s channel as the setup we consider

in this section and then analyze achievable distortion regions for several different cases.

The rest of the paper is organized as follows. In section II, we present some prior works which are closely related to ours.

The outer bound is given in section III and two proposed schemes are given in section IV. In section V, we analyze the

performance of the proposed schemes under SNR mismatch. These proposed schemes are then extended to the generalized

cognitive radio channels in section VI. Some conclusions are given in VII.

ΛV S=

σ2

V

ρσVσS

σ2

ρσVσS

S

?

.

(5)

II. RELATED WORKS ON JSCC WITH INTERFERENCE KNOWN AT TRANSMITTER

In [7], Lapidoth et al. consider the 2×1 multiple access channel in which two transmitters wish to communicate their sources,

which are drawn from a bi-variate Gaussian distribution, to a receiver which is interested in reconstructing both sources. There

are some similarities between the work in [7] and here. However, an important difference is that the transmitters are not allowed

to cooperate with each other, i.e., for the particular transmitter, the interference is not known.

In [8], Tian et al. consider transmitting a bi-variate Gaussian source over 1×2 Gaussian Broadcast Channel. In their setup,

the source consisting of two components Vn

receiver is only interested in one part of the sources. They proposed a HDA scheme which performs optimally in terms of

distortion region under all SNRs. At first glance, this problem is again similar to ours if we ignore receiver 2 and focus on

the other. Then this problem reduces to communicating Vn

crucial difference is that this side-information does not appear in the received signal.

Joint source-channel coding for point to point communications over Gaussian channels has been widely discussed. e.g. [3],

[9], [10]. However, they either don’t consider interference ([9], [10]) or assume independence of source and interference ([3]).

In [3], Wilson et al. proposed a HDA coding scheme for the typical writing on dirty paper problem in which the source is

independent of the interference. This HDA scheme is originally proposed to perform well in the case of a SNR mismatch. In

[3], the authors showed that their HDA scheme not only achieves the optimal distortion in the absence of SNR mismatch but

also provides gracefully degradation in the presence of SNR mismatch. In the following sections, we will discuss this scheme

in detail and then propose a coding scheme based on this one.

From now on, since all the random variables we consider are i.i.d. in time, i.e. V (i) is independent of V (j) for i ?= j, we

will drop the index i for the sake of convenience.

1 and Vn

2 memoryless and stationary bi-variate Gaussian distributed and each

1 with correlated side-information Vn

2 given at the transmitter. A

III. OUTER BOUNDS

A. Outer Bound 1

For comparison, we first present a genie-aided outer bound. This outer bound is derived in a similar way to the one in [11]

in which we assume that S is revealed to the decoder by a genie. Thus, we have

1

2logσ2

V(1 − ρ2)

Dob

(a)

≤ I(V ;?V |S)

≤ I(V ;Y |S)

= h(Y |S) − h(Y |S,V )

= h(X + Z|S) − h(Z)

(b)

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2

(c)

≤ h(X + Z) − h(Z)

(d)

≤1

2log

?

1 +P

N

?

,

(6)

where (a) follows from the rate-distortion theory [1], (b) is from the data processing inequality, (c) is due from that conditioning

reduces differential entropy and (d) comes from the fact that Gaussian density maximizes the differential entropy. Therefore,

we have the outer bound as

Dob,1=σ2

1 + P/N

V(1 − ρ2)

.

(7)

Note that this outer bound is in general not tight for our setup since in the presence of correlation, giving S to the decoder

also offers a correlated version of the source that we wish to estimate. For example, in the case of ρ = 1, giving S to the

decoder implies that the outer bound is Dob= 0 no matter what the received signal Y is. On the other hand, if ρ = 0, the

setup reduces to the one with uncorrelated interference and we know that this outer bound is tight. Now, we present another

outer bound that improves this outer bound for some values of ρ.

B. Outer Bound 2

Since S and V are drawn from a jointly Gaussian distribution with covariance matrix given in (5), we can write

S = ρσS

σV

V + Nρ,

(8)

where Nρ∼ N?0,(1 − ρ2)σ2

S

?and is independent to V . Now, suppose a genie reveals only Nρto the decoder, we have

1

2logDob,2

Dob

σ2

V

=1

2logvar(V |Nρ)

(a)

≤ I(V ;?V |Nρ)

≤ I(V ;Y |Nρ)

= h(Y |Nρ) − h(Y |Nρ,V )

= h(X + ρσS

σV

(c)

≤ h(X + ρσS

σV

?

(b)

V + Z|Nρ) − h(Z)

V + Z) − h(Z)

?

N

1 +(√P + ρ?σ2

(d)

≤1

2log

var

X + ρσS

σVV + Z

?

(e)

≤1

2log

S)2

N

?

,

(9)

where (a)-(d) follow from the same reasons with those in the previous outer bound and (e) is due from the Cauchy-Schwartz

inequality that states that the maximum occurs when X and V are collinear. Thus, we have

Dob,2=

σ2

V

1 + (√P + ρ?σ2

S)2/N.

(10)

Note that although the encoder knows the interference S exactly instead of just Nρ, the outer bound is valid since S is a

function of V and Nρ.

Remark 1: If ρ = 0, this outer bound reduces to the previous one and is tight. If ρ = 1, the genie actually reveals nothing

to the decoder and the setup reduces to the one considered in [12] that the encoder is interested in revealing the interference

to the decoder. For this case, we know that this outer bound is tight. However, this outer bound is in general optimistic except

for two extremes. It is due to the fact that in derivations, we assume that we can simultaneously ignore the Nρand use all the

power to take advantage of the coherent part. Despite this, the outer bound still provides an insight that in order to build a

good coding scheme that one should try to use a portion of power to make use of the correlation and then use the remaining

power to avoid Nρ.

Further, it is natural to combine these two outer bounds as

Dob= max{Dob,1,Dob,2}.

(11)

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IV. PROPOSED SCHEMES

A. Uncoded Scheme

We first analyze the distortion of the uncoded scheme where the transmitted signal is simply the scaled version of the source

?

Thus, (2) becomes

?

σ2

V

X =

P

σ2

V

V.

(12)

Y =

P

V + S + Z.

(13)

The receiver forms the linear MMSE estimate of V from Y as?V = βY , where

β =

σ2

V(?P/σ2

V+ ρσS/σV)

P + σ2

S+ N + 2?P/σ2

?

VρσVσS

.

(14)

The corresponding distortion is then given as

Dunc= σ2

V

1 − β(

?

P

σ2

V

+ ρσS

σV

)

?

.

(15)

Remark 2: If ρ = 1 and σ2

transmitting V over an AWGN channel Z with power constraint (√P +?σ2

channel state S to the receiver. In [12], the authors have shown that the pure amplification (uncoded) scheme is optimal for

this problem. Therefore, we can expect that the uncoded scheme will eventually achieve the optimal distortion when ρ = 1.

V= σ2

S, the source and the interference are exactly the same and the problem reduces to

V)2. From [13] [14], we know that the uncoded

scheme is optimal for this case. One can also think of this scenario as that the transmitter is only interested in revealing the

B. Naive DPC Scheme

Another existing scheme is the concatenation of a optimal source code and a DPC. The optimal source code quantizes the

analog source with a rate arbitrarily close to the channel capacity 1/2log(1 + P/N). Then, the DPC ignores the correlation

between the source and interference (this can be done by a randomization and de-randomization pair) and encodes the

quantization output accordingly. Since the DPC achieves the rate equal to that when there is no interference at all, the receiver

can correctly decode these digital bits with high probability. By the rate-distortion theory, we have the corresponding distortion

as

DDPC=

σ2

V

1 + P/N.

(16)

Remark 3: In the absence of correlation, i.e., ρ = 0, the problem reduces to the typical writing on dirty paper setup and it

is known that this scheme is optimal but the uncoded scheme is strictly suboptimal. Therefore, we can expect that when the

correlation is small, this naive DPC scheme will outperform the uncoded scheme.

C. Superposition-Based Scheme with Digital DPC

We now propose a superposition-based scheme which retains the advantages of the above two schemes. This scheme can

be regarded as an extended version of the coding scheme in [10] to the setup we consider. As shown in Fig. 2, the transmitted

signal of this scheme is the superposition of the analog part Xawith power Paand the digital part Xdwith power P − Pa.

The motivation here is to allocate some power for the analog part to make use of the interference which is somewhat coherent

to the source for large ρ’s and to assign more power to the digital part to avoid the interference when ρ is small. The analog

part is the scaled version of linear combination of source and interference as

Xa=√a(γV + (1 − γ)S),

where Pa∈ [0,P], a = Pa/σ2

σ2

(17)

a, γ ∈ [0,1] and

a= γ2σ2

V+ (1 − γ)2σ2

S+ 2γ(1 − γ)ρσVσS.

(18)

The received signal is given by

Y = Xd+ Xa+ S + Z

= Xd+√a(γV + (1 − γ)S) + S + Z

= Xd+√aγV +?1 +√a(1 − γ)?S + Z

= Xd+ S′+ Z,

(19)

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4

+

S

W.-Z.

γ

d

X

X

V

Costa

T

V

1 γ

−

2

a

a

P

σ

a

X

Fig. 2. Superposition-based scheme.

where Xdis chosen to be orthogonal to S and V . The receiver first makes an estimate from Y only as V′= βY with

√a(γσ2

V+ (1 − γ)ρσVσS) + ρσVσS

P + N + σ2

The corresponding MSE is

?

Thus, we can write V = V′+ W with W ∼ N(0,D∗).

We now refine the estimate through the digital part, which is the concatenation of a Wyner-Ziv coding and a DPC. Since

the DPC achieves the rate equal to that when there is no interference at all, the encoder can use the remaining power P −Pa

to reliably transmit the refining bits T with a rate arbitrarily close to

?

The resulting distortion after refinement is then given as

β =

S+ 2√a((1 − γ)σ2

?√a(γ + (1 − γ)ρσS

S+ γρσVσS).

(20)

D∗= σ2

V

1 − β

σV

) + ρσS

σV

??

.

(21)

R =1

2log

1 +P − Pa

N

?

.

(22)

Dsep= inf

γ, Pa

D∗

1 +P−Pa

N

.

(23)

In Appendix A, for self-containedness, we briefly summarize the digital Wyner-Ziv scheme to illustrate how to achieve the

above distortion.

It is worth noting that setting γ = 1 gives us the lowest distortion always. i.e., super-imposing S onto the transmitted signal

is completely unnecessary. However, it is in general not true for the cognitive radio setup. We will discuss this in detail in

section VI.

Remark 4: Different from the setup considered in [10] that the optimal distortion can be achieved by any power allocation

between coded and uncoded transmissions, in our setup the optimal distortion is in general achieved by a particular power

allocation which is a function of ρ. For example, in the absence of correlation, i.e., S is completely independent to V , one

can simply set Pa= 0 and this scheme reduces to the naive DPC which is optimal in this case. On the other hand, if ρ = 1,

the optimal distortion is achieved by setting Pa= P. Moreover, for ρ > 0, it is beneficial to have a non-zero Pamaking use

of the correlation between the source and the interference.

D. HDA Scheme

Now, let us focus on the HDA scheme shown in Fig. 3 obtained by replacing the digital part in Fig. 2 by the HDA scheme

given in [3]. The analog signal remains the same as (17) and the HDA output is referred to as Xh. Therefore, we have

Y = Xh+√aγV +?1 +√a(1 − γ)?S + Z

Again, the HDA scheme regards S′as interference and V′described previously as side-information. The encoding and decoding

procedures are similar to that in [3] but the coefficients need to be re-derived to fit our setup (the reader is referred to [3] for

details).

Let the auxiliary random variable U be

U = Xh+ αS′+ κV,

= Xh+ S′+ Z.

(24)

(25)

where Xh∼ N(0,Ph) independent to S′and V and Ph= P − Pa. The covariance matrix of S′and V can be computed by

(5).

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5

+

S

γ

h

X

X

V

HDA

V

1 γ

−

2

a

a

P

σ

a

X

Fig. 3.HDA scheme.

Codebook Generation: Generate a random i.i.d. codebook U with 2nR1codewords, reveal this codebook to both transmitter

and receiver.

Encoding: Given realizations s′and v, find a u ∈ U such that (s′,v,u) is jointly typical. If such an u can be found, transmit

xh= u − αs′− κv. Otherwise, an encoding failure is declared.

Decoding: The decoder looks for a ˆ u such that (y,v′, ˆ u) is jointly typical. A decoding failure is declared if none or more

than one such ˆ u are found. It is shown in [3] that if n → ∞ and the condition described later is satisfied, the probability of

ˆ u ?= u → 0.

Estimation: After decoding u, the receiver forms a linear MMSE estimate of v from y and u. The distortion is then obtained

as

Dhda= inf

γ, Pa

?σ2

V− ΓTΛ−1

UYΓ?,

(26)

where ΛUY is the covariance matrix of U and Y , and

Γ = [E[V U],E[V Y ]]T.

(27)

In the encoding step, to make sure the probability of encoding failure vanishes with increasing n, we require

R1> I(U;S′,V )

= h(U) − h(U|S′,V )

= h(U) − h(Xh+ αS′+ κV |S′,V )

(a)

= h(U) − h(Xh)

=1

Ph

2logE[U2]

.

(28)

where (a) follows because Xhis independent of S′and V .

Further, to guarantee the decodability of U in the decoding step, one requires

R1< I(U;Y,V′)

= h(U) − h(U|Y,V′)

= h(U) − h(U − αY − κV′|Y,V′)

(a)

= h(U) − h(κW + (1 − α)Xh− αZ|Y ),

(29)

where (a) follows from V′= βY . By choosing

α =

Ph

Ph+ N

(30)

and

κ2=

P2

h

(Ph+ N)D∗,

(31)

one can verify that (28) and (29) are satisfied. Note that in (28) what we really need is R1≥ I(U;S′,V ) + ε and in (29) it

is R1≤ I(U;Y,V′)−δ. However, since ε and δ can be made arbitrarily small, these are omitted for the sake of convenience

and to maintain clarity.

Remark 5: It can be verified that the distortions in (23) and (26) are exactly the same. However, it has been shown in [3]

that the HDA scheme can provide graceful degradation in the SNR mismatch case.

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6

012345

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

SNR (dB)

−10log10(D)

Uncoded

Naive DPC

Superposition−based

HDA

Outer bound

Fig. 4.

P

Nvs D, ρ = 0.3.

00.2 0.40.60.81

8

9

10

11

12

13

14

ρ

−10log10(D)

Uncoded

Naive DPC

Superposition−based

HDA

Outer bound 1

Outer bound 2

Fig. 5.

ρ vs D with σV = σS= 1 and

P

N= 10.

E. Numerical Results

In Fig. 4, we plot the distortion (in −10log10(D)) for coding schemes and outer bounds described above as a function of SNR.

In this figure, we set σ2

Moreover, for this case, these two schemes not only outperform others but also approach the outer bound (maximum of two)

very well.

We then fix the SNR and plot the distortion as a function of ρ in Fig. 5. The parameters are set to be σ2

P = 10, and N = 1. As we discussed in Remark 2 and Remark 3, the naive DPC scheme performs optimally when ρ = 0 and

performs better than the uncoded scheme at small ρ regime. However, the uncoded scheme outperforms the naive DPC scheme

at large ρ regime and eventually achieves optimum when ρ = 1. Further, it can be seen that both the proposed schemes exactly

the same and the achievable distortion region with the proposed scheme is larger than what is achievable with the naive DPC

scheme and the uncoded scheme. It can be observed that although the proposed schemes perform close to the outer bound

over a wide range of ρs, the outer bound and the inner bound do not coincide however, leaving room for improvement either

of the outer bound or the schemes.

V= σ2

S= 1 and ρ = 0.3. As we expected, two proposed schemes have exactly the same performance.

V= σ2

S= 1,

V. PERFORMANCE ANALYSIS IN THE PRESENCE OF SNR MISMATCH

In this section, we study the distortions for the proposed schemes in the presence of SNR mismatch i.e., we consider

the scenario where instead of knowing the exact channel SNR, the transmitter only knows a lower bound of channel SNR.

Specifically, we assume that the actual channel noise to be Za∼ N(0,Na) but the transmitter only knows that Na≤ N so

that it designs the coefficients for this N. In what follows, we analyze the performance for both proposed schemes under the

above assumption.

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A. Superposition-Based Scheme with Digital DPC

Since the transmitter designs its coefficients for N, it aims to achieve the distortion Dsep given in (23). It first quantizes

the source to T by a Wyner-Ziv coding with side-information D∗given in (21) and then encodes the quantization output by

a DPC with a rate

R =1

2log

?

1 +P −˜Pa

N

?

,

(32)

where˜Pais the power allotted to Xasuch that the distortion in the absence of SNR mismatch is minimized. i.e.,

˜Pa= arginf

Pa

D∗

1 +P−Pa

N

.

(33)

At receiver, since Na≤ N, the DPC decoder can correctly decode T with high probability. Moreover, the receiver forms

the MMSE of V from Y as V′

√a(γσ2

V+ (1 − γ)ρσVσS) + ρσVσS

P + Na+ σ2

?

Thus, the problem reduces to the Wyner-Ziv problem with mismatch side-information. In Appendix B, we show that for this

problem, one can achieve

a= βaY with

βa=

S+ 2√a((1 − γ)σ2

?√a(γ + (1 − γ)ρσS

S+ γρσVσS),

(34)

D∗

a= σ2

V

1 − βa

σV

) + ρσS

σV

??

.

(35)

Dsep,mis=

D∗D∗

a

D∗D∗

a+ (D∗− D∗

a)DsepDsep.

(36)

Unlike the typical separation-based scheme that we have seen in [3], the proposed superposition-based scheme (whose

digital part can be regarded as a separation-based scheme) can still take advantage of better channels through mismatched

side-information. i.e., this scheme does not suffer from the pronounced ”threshold effect”.

B. HDA Scheme

Although it is shown in Appendix B that the performance of the HDA scheme is exactly the same with the digital Wyner-Ziv

scheme under side-information mismatch, this problem with HDA scheme cannot be reduced to the Wyner-Ziv problem with

mismatch side-information as we did for the superposition-based scheme. It is due from that the HDA scheme still makes an

estimate of V from U which is a function of S. Fortunately, as shown in [3], the HDA scheme is capable of making use of

SNR mismatch.

Similar to the superposition-based scheme, we design the coefficients for channel parameter N. The HDA scheme regards

D∗as side-information and S′as interference. It generates the auxiliary random variable U given by (25) with coefficients

described by (30) and (31). Since Na≤ N, the receiver can correctly decode U with high probability. The receiver then forms

the MMSE as described in (26) and (27). Note that E[Y2] in ΛUY should be modified appropriately to address the fact that

the actual noise variance is Nain this case.

Remark 6: In [12], the optimal tradeoff between the achievable rate and the error in estimating the interference at the

designed SNR is studied. In [3], the authors also studied a somewhat similar problem. They compare the distortions of the

digital scheme and the HDA scheme in estimating the source V and the interference S as we move away from the designed

SNR. One important observation is that the HDA scheme outperforms the separation-based scheme in estimating the source;

however, the separation-based scheme is better than the HDA scheme if one is interested in estimating the interference. Here,

since the effective interference S′includes the uncoded signal√aV in part and the source is assumed to be correlated to the

interference, estimating the source V is equivalent to estimating a part of S′. Thus, one can expect that if the Pawe choose

and the correlation ρ are large enough, the benefit coming from using the HDA scheme to estimate the source may be less

than that from adopting the superposition-based scheme to estimate a part of S′. Consequently, for a sufficiently large Paand

ρ, the superposition-based scheme may be better than the HDA scheme in the presence of SNR mismatch.

C. Numerical Results

Now, we compare the performance of the above two schemes and the scheme that knows the actual SNR. The parameters

are set to be σ2

large (ρ = 0.5) correlations. Two examples for designed SNR = 0 dB and 10 dB are given in Fig. 6 and Fig. 7, respectively.

In Fig. 6, we consider the case that the designed SNR is 0 dB which is relatively small compared to the variance of

interference. For this case, we can see that which scheme performs better in the presence of SNR mismatch really depends

on ρ. It can be explained by the observations made in Remark 6 and the power allocation strategy. For this case the optimal

power allocation˜Pais proportional to ρ. For ρ = 0.1 case, since the correlation is small and the assigned˜Pais also small, the

V= σ2

S= 1. We plot the −10log10(D) as we move away from the designed SNR for both small (ρ = 0.1) and

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012345

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

SNR (dB)

−10log10(D)

Superposition−based (ρ = 0.1)

HDA (ρ = 0.1)

Actual SNR (ρ = 0.1)

Superposition−based (ρ = 0.5)

HDA (ρ = 0.5)

Actual SNR (ρ = 0.5)

Fig. 6.SNR mismatch case for small designed SNR.

1015 20 2530 35 40

10

15

20

25

30

35

40

SNR (dB)

−10log10(D)

Superposition−based (ρ = 0.1)

HDA (ρ = 0.1)

Actual SNR (ρ = 0.1)

Superposition−based (ρ = 0.5)

HDA (ρ = 0.5)

Actual SNR (ρ = 0.5)

Fig. 7.SNR mismatch case for large designed SNR.

HDA scheme is better than the superposition-based scheme. On the other hand, for ρ = 0.5 case, we allot a relatively large

power to˜Paso that one may get a better estimate if we try to use the superposition-based scheme to estimate a part of S′.

This property is further discussed in the Appendix C.

In Fig. 7, we design the coefficients for SNR = 10 dB which can be regarded as relatively large SNR compared to the

variance of interference. For this case, the optimal power allocation˜Pa for both ρ = 0.1 and ρ = 0.5 are relatively small.

Therefore, the performance improvement provided by the HDA scheme is larger than that provided by the superposition-based

scheme for both cases.

In Fig. 8, we plot the proposed schemes with different choices of Paunder the same channel parameters with those in the

previous figure for ρ = 0.1. We observe that for both schemes, if we compromise the optimality at the designed SNR, it is

possible to get better slopes of distortion than that obtained by setting Pa=˜Pa. In other words, we can obtain a family of

achievable distortion under SNR mismatch by choosing Pa∈ [0,P].

VI. JSCC FOR GENERALIZED COGNITIVE RADIO CHANNELS

There has been a lot of interest in cognitive radio since it has been proposed in [15] for flexible communication devices and

higher spectral efficiency. In a conventional cognitive radio setup, the lower priority user (usually referred to as the secondary

user) listens to the wireless channel and transmits the signal only through the spectrum not used by the higher priority user

(referred to as the primary user).

In [5], Devroye et al. studied the generalized cognitive radio channels in which simultaneous transmission over the same

time or frequency is allowed. This channel can be modeled as a typical two-user interference channel except that one of users

knows exactly what the other plans to transmit.The authors then provide inner and outer bounds on how much rate two users

Page 10

9

012345

3.5

4

4.5

5

5.5

6

6.5

7

7.5

SNR (dB)

−10log10(D)

Actual SNR

HDA (Pa = Pa,opt)

HDA (Pa = 0.9)

Superposition−based (Pa = Pa,opt)

Superposition−based (Pa = 0.9)

Fig. 8. Proposed schemes with different choices of Pa.

can transmit simultaneously for such generalized cognitive radio channel. Their achievable scheme is based on the DPC and

the Han-Kobayashi scheme [16].

In this section, we consider the same generalized cognitive radio channels as in [5] and focus on the case when both two

users have analog information V1and V2. We are interested in the distortion region which describes how much distortion two

users can achieve simultaneously. In particular, we consider the case that two sources are correlated with a covariance matrix

given by

?

As we mentioned before, we first look at the distortion of the secondary user only and regard it as the setup in section II.

An achievable distortion region is obtained by forcing the primary user to use the uncoded scheme and using the proposed

schemes given in section IV for the secondary user. In fact, since the primary user does not have any side-information, the

analog transmission seems to be an optimal choice. Further notice that since we do not consider SNR mismatch here, it makes

no difference which proposed schemes we use.

In what follows, we show that when the correlation is large, adopting the proposed scheme at the secondary user not only

takes advantage of this correlation but also benefits the primary user. On the other hand, when ρ is small, the proposed scheme

helps the secondary user to avoid the interference introduced by the primary user.

As shown in Fig. 9, in a generalized cognitive radio channel, two users wish to transmit their own sources to the corresponding

receiver through an interference channel with direct channel gain 1 and cross channels h1and h2representing the real-valued

channel gains from user 1 to user 2 and vice versa, respectively. The power constraints imposed on the outputs of user 1 and

2 are P1and P2, respectively. Different from interference channels, in cognitive radio channels, we assume that the secondary

user knows V1non-causally. Here, we also assume that the channel coefficient h1is known by the secondary user. The received

signals are given by

?

where Zi∼ N(0,Ni) for i ∈ {1,2}.

Let the primary user simply transmit the scaled version of the uncoded source

?

ΛV1V2=

σ2

V1

ρσV1σV2

σ2

V2

ρσV1σV2

?

.

(37)

Y1

Y2

?

=

?

1h1

1h2

??

X1

X2

?

+

?

Z1

Z2

?

.

(38)

X1=

P1

σ2

V1

V1.

(39)

Therefore, the bottom channel in Fig. 9 reduces to the situation we considered in the previous section with source V = V2

and interference S = h1X1. The covariance matrix becomes (5) with

σ2

σ2

V= σ2

S= h2

V2,

(40)

(41)

1P1.

The secondary user then encodes its source to X2by the HDA scheme described previously in section IV-D with power P2

and coefficients according to (30) and (31). With these coefficients, the corresponding distortion D2is computed by (26) and

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10

ENC 2

+

DEC 2

2

V

2ˆV

2

Z

2

X

2Y

ENC 1

+

DEC 1

1ˆV

1 Z

1

X

1Y

1

1

1h

2h

Primary user

Secondary user

1V

Fig. 9. Setup of cognitive radio channels.

(27). At the receiver 1, the received signal is

Y1= X1+ h2X2+ Z1

=?1 + (1 − γ)√ah1h2

?X1+ h2Xh+ h2

√aγV2+ Z1.

(42)

The decoder 1 then forms a linear MMSE estimate from Y1given by?V1= βY1, where

β =E[V1Y1]

E[Y2

1]

(43)

with

E[V1Y1] =?1 + (1 − γ)√ah1h2

E[Y2

2Ph+ 2√ah2γρ

??

P1σ2

V1+ h2

√aγρσV1σV2

(44)

1] =?1 + (1 − γ)√ah1h2

h2

?2P1+ ah2

P1σ2

V2

2γ2σ2

V2+

?

?1 + (1 − γ)√ah1h2

V1− βE[V1Y1]

?.

(45)

Therefore, the corresponding distortion is

D1= σ2

(46)

It can be verified that assigning γ = 1 leads to a suboptimal D1in general. Thus, as we mentioned before, one may want

to assign a non-zero power to transmit S in order to achieve a larger distortion region.

We can then optimize the power allocation for particular performance criteria. For instance, if one desires achieving the

minimum distortion for the secondary user, γ should be set to be 1. However, if the target is to obtain the largest achievable

distortion region under a total power constraint P1+ P2= P, one should optimize over P1∈ [0,P], Pa∈ [0,1 − P1], and

γ ∈ [0,1]. We briefly discuss these examples below.

1. Greedy Case: We first consider the greedy case where the secondary user focus on reducing its own distortion. As we

mentioned before, the proposed scheme should always set γ = 1 for this case. For comparison, an outer bound on distortion

region for this case is given as follows. Suppose that there is a genie that reveals V1 to the decoder 2 and V2 to both the

encoder 1 and the decoder 1. Similar to the derivation in section III, one obtains

D1ob=σ2

V1(1 − ρ2)

1 + P1/N1

V2(1 − ρ2)

1 + P2/N2

.

(47)

D2ob=σ2

.

(48)

From now on, we only present the outer bound 1 since in the numerical results we consider in the following, this outer bound

is tighter than the outer bound 2. However, one can also derive the outer bound 2 for these cases and take the maximum of

two by a similar way given in section III.

Numerical examples are given in Fig. 10 and 11, in which we set σ2

total power P = 2. The correlation between sources are ρ = 0 and ρ = 0.3, respectively. In both examples, we do not perform

optimization over Ph and Pa with respect to particular criteria. Instead, we plot many choices of Ph and Pa which satisfy

P2= Ph+ Pa.

In Fig. 10, we observe that the proposed scheme achieves the outer bound at two corners in the absence of correlation. The

left corner point can be achieved by assigning P2= P and the right corner point can be achieved by setting P1= P. For

V1= σ2

V2= N1= N2= 1, h1= h2= 0.5, and the

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0.4 0.5 0.60.70.80.91

0.4

0.5

0.6

0.7

0.8

0.9

1

D2

D1

P = 2, N1 = N2 = 1, ρ = 0

outer bound

Fig. 10. Greedy case, ρ = 0.

0.4 0.5 0.60.7 0.8 0.91

0.4

0.5

0.6

0.7

0.8

0.9

1

D2

D1

P = 2, N1 = N2 = 1, ρ = 0.3

outer bound

Fig. 11. Greedy case, ρ = 0.3.

other points, the inner and outer bounds do not coincide. This may be due from that in deriving the outer bound, the genie

reveals to the primary user too much information so that the outer bound may not be tight (recall that for the ρ = 0 case, the

outer bound for the secondary user is tight). Despite this, the inner bound is close to the outer bound. In Fig. 11, we give an

example where ρ = 0.3. One can observe that compared to the result in Fig. 10, the correlation helps both users in terms of

distortion. And again, although the outer bound is not tight, the gap is reasonably small.

2. Non-Greedy Case: We now consider the case that the secondary user is willing to help the primary user. i.e., the γ ∈ [0,1].

For this case, the outer bounds must be modified to address the fact that the secondary user uses a part of its power to transmit

V1. For the primary user, suppose there is a genie that reveals V2and the HDA encoder to both encoder 1 and decoder 1, i.e.,

Xhis also known at both sides. We have

n

2logσ2

= h(X1+ h1X2+ Z1|V2,Xh) − h(Z1)

≤ h??1 + (1 − γ)√ah1h2

2log(1 + snr1),

V1(1 − ρ2)

D1ob

≤ I(V1;?V1|V2) ≤ I(V1;Y1|V2)

?X1+ Z1)?− h(Z1)

=n

(49)

where

snr1=P1(1 + (1 − γ)√ah1h2)2

N1

.

(50)

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0.4 0.50.60.70.80.91

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

D2

D1

P = 2, N1 = N2 = 1, ρ = 0

outer bound

Fig. 12.Non-Greedy case, ρ = 0.

Similarly, we assume a genie gives away V1to decoder 2 so that we have

2logσ2

= h(X2+ h2X1+ Z2|V1) − h(Z2)

≤ h?Xh+ γ√aV2+ Z2)?− h(Z2)

n

V2(1 − ρ2)

D2ob

≤ I(V2;?V2|V1) ≤ I(V2;Y2|V1)

=n

2log(1 + snr2),

(51)

where

snr2=Ph+ aγ2σ2

V2

N2

.

(52)

Thus, for each choice of P1, Pa, and γ we have the outer bound as

˜D1ob=σ2

V1(1 − ρ2)

1 + snr1

V2(1 − ρ2)

1 + snr2

(53)

˜D2ob=σ2

.

(54)

The outer bound of this case is obtained numerically by taking the lower convex envelope over all

The numerical results for ρ = 0 and ρ = 0.3 are given in Fig. 12 and Fig. 13, respectively. In both figures, all the parameters

are set to be the same as those in the previous two examples. We observe that if the secondary user is willing to help the

primary user, the achievable distortion region is larger than that of greedy case.

3. Coexistence Conditions: In [6], the coexistence conditions are introduced to understand the system-wise benefits of

cognitive radio. The authors study the largest rate that the cognitive radio can achieve under these coexistence constraint

described as follows,

1. the presence of cognitive radio should not create rate degradation for the primary user, and

2. the primary user does not need to use a more sophisticated decoder than it would use in the absence of the cognitive

radio. i.e, a single-user decoder is enough.

Similar to this idea, we study the distortion of the secondary user under the modified coexistence constraint as

1. the presence of cognitive radio should not increase distortion for the primary user, and

2. the primary user uses a single-user decoder.

Let the power constraints be P1 and P2 for the primary and the secondary user, respectively, and P1+ P2 = P. In the

absence of the cognitive radio, the distortion of the primary user is

?˜D1ob,˜D2ob

?

.

D∗

1=

σ2

V1

1 + P1/N1.

(55)

The outer bound on the secondary user under the coexistence conditions is given as

Dcoexist,ob= inf

Pa, γ,˜ D1ob≤D∗

1

˜D2ob,

(56)

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13

0.40.5 0.60.70.80.91

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

D2

D1

P = 2, N1 = N2 = 1, ρ = 0.3

outer bound

Fig. 13.Non-Greedy case, ρ = 0.3.

00.511.52

0.4

0.5

0.6

0.7

0.8

0.9

1

P2

D2

proposed scheme (ρ = 0)

outer bound (ρ = 0)

proposed scheme (ρ = 0.3)

outer bound (ρ = 0.3)

Fig. 14.Coexistence case, ρ = 0 and ρ = 0.3.

where˜D1oband˜D2ob are given by (53) and (54), respectively. An example is shown in Fig. 14. All the parameters in this

figure are the same with those in Fig. 10-13.

VII. CONCLUSIONS

In this paper, we have discussed the joint source-channel coding problem with interference known at the transmitter. In

particular, we considered the case that the source and the interference are correlated with each other. According to the

observations on the uncoded scheme and the naive DPC scheme, we proposed a superposition-based scheme with digital

DPC and a HDA scheme which can adapt with ρ. The performance of these two schemes under SNR mismatch are also

discussed. Different from typical separation-based schemes suffering from the pronounced threshold effect in the presence of

SNR mismatch, both the proposed schemes can benefit from a better side-information acquired at the decoder and thus, provide

a graceful degradation under SNR mismatch. However, there is a difference between the performance of the two proposed

schemes under a SNR mismatch and which scheme is better depends on the designed SNR and ρ.

These two schemes are then applied to cognitive radio channels and achievable distortion regions are discussed for different

cases. To the best of our knowledge, this is the first joint source-channel coding scheme for cognitive radio channels. We have

also provided outer bounds on these distortion regions. Despite the fact that the outer bounds are not tight in general, the

numerical results have shown that the gap between the inner bound and the outer bound is reasonably small.

Page 15

14

APPENDIX A

DIGITAL WYNER-ZIV SCHEME

In this appendix, we summarize the digital Wyner-Ziv scheme for lossy source coding with side-information V′(V = V′+W

with W ∼ N(0,D∗)) at receiver. Similar to the previous sections, we omit all the ε and/or δ intentionally for the sake of

convenience and to maintain clarity.

Suppose the side-information is available at both sides, the least required rate RWZ for achieving a desired distortion D is

[4]

RWZ=1

2logD∗

D.

(57)

Let us set this rate to be arbitrarily close to the rate given in (22), the rate that the channel can support with arbitrarily small

error probability. The best possible distortion one can achieve for this setup is then given as

D =

D∗

1 +P−Pa

N

.

(58)

This distortion can be achieved as follows [4],

1. Let T be the auxiliary random variable given by

T = αsepV + B,

(59)

where

αsep=

?

D∗− D

D∗

(60)

and B ∼ N(0,D). Generates a length n i.i.d. Gaussian codebook T of size 2nI(T;V )and randomly assign the codewords into

2nRbins with R chosen from (22). For each source realization v, find a codeword t ∈ T such that (v,u) is jointly typical.

If none or more than one are found, an encoding failure is declared.

2. For each chosen codeword, the encoder transmit the bin index of this codeword by the DPC with rate given in (22).

3. The decoder first decodes the bin index (the decodability is guaranteed by the rate we chose) and then looks for a

codewordˆt in this bin such that (ˆt,v′) is jointly typical. If this is not found, a dummy codeword is selected. Note that as

n → ∞, the probability thatˆt ?= t vanishes. Therefore, we can assume thatˆt = t from now on.

4. Finally, the decoder forms the MMSE from t and v′as ˆ v = v′+ ˆ w with

αsepD∗

α2

It can be verified that for the choice of α the required rate is equal to (57) and the corresponding distortion are

ˆ w =

sepD∗+ D(t − αsepv′).

(61)

E[(V −?V )2] = E[(W −?

= D∗

1 −

α2

W)2]

?

?

α2

sepD∗

sepD∗+ D

= D.

(62)

APPENDIX B

WYNER-ZIV WITH MISMATCHED SIDE-INFORMATION

In this appendix, we calculate the expected distortion of the digital Wyner-Ziv scheme in the presence of side-information

mismatch. Specifically, we consider the Wyner-Ziv problem with an i.i.d. Gaussian source and the MSE distortion measure.

Let us assume that the best achievable distortion in the absence of side-information mismatch to be D. The encoder believes

that the side-information is V′, and V = V′+ W with W ∼ N(0,D∗). However, the side-information turns out to be V′

and has the relation V = V′

suffered by the decoder .

Since the encoder has been fixed to deal with the side-information, V′, at decoder, the auxiliary random variable is as in

(59) with the coefficient given in (60).

Since the decoder knows the actual side-information, V′

the MMSE estimate?

Wa=

α2

sepD∗

a

a+ Wawith Wa∼ N(0,D∗

a). Under the same rate, we want to calculate the actual distortion Da

a, perfectly, it only has to estimate Wa. By the orthogonality principle,

Wacan be obtained as

?

αsepD∗

a

a+ D(T − αsepV′

a)

(63)

Therefore, the estimate of the source is?V = V′

a+?

=

Wa. The corresponding distortion is given as

Da= E[(V −?V )2] = E[(Wa−?

D∗D∗

a+ (D∗− D∗

Wa)2]

D∗D∗

a

a)DD

(64)